I don't know anything about the case not
covered by Theorem REF : can a (finitely presented) irreducible lattice in
\(G=\prod _{i \in I} {\mathbf {G}}_i(k_i)\)  with \(\textnormal {rk}\,G>1\)  and all \(k_i\)  of (necessarily
the same, see Theorem REF ) positive characteristic, be
a Kähler group? This question already appeared in [1]}.
